List of practice Questions

Evaluate the infinite series: \[ \left|\begin{array}{ccc} 2 & 1 & \frac{1}{3} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{3} & \frac{1}{2} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{9} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{27} \\ 3 & 1 & 1 \\ \end{array}\right| + \cdots = ? \]

The range of the real valued function \( f(x) = \cos^{-1} \left( \dfrac{3}{\sqrt{9x^2 - 12x + 22}} \right) \) is

If a function \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{\sqrt{1+ax^2+bx^3}-\sqrt{1-ax^2-bx^3}}{x^2}, & x<0 \\ 5, & x = 0 \\ \frac{\tan3x-\sin3x}{bx^3}, & x>0 \end{cases} \] and is continuous at \( x = 0 \), then the geometric mean of \( a \) and \( b \) is:

Match the following polymers with their applications:

List-I (Polymer)	List-II (Used in making)
Polymer 1	Application 1
Polymer 2	Application 2
Polymer 3	Application 3
Polymer 4	Application 4

The structure of the product 'Z' in the reaction sequence is: \[ \text{CHO} \] \[ \mid \] \[

The amphoteric oxide of Vanadium (V) reacts with alkali and forms an oxoion \( X \) and with acid forms an oxoion \( Y \). The oxidation states of \( V \) in \( X \) and \( Y \) are respectively:

The adsorption of a gas on a solid surface follows Freundlich adsorption isotherm. At \( T \)(K), the gas pressure is \( 2 \) atm. What is the value of \( \frac{x}{m} \)? (\( n = 2 \) and \( k \) = constant)

Which one of the following acts as an autocatalyst during titration of \( KMnO_4 \) and oxalic acid in presence of dilute \( H_2SO_4 \)?

The correct order of density of Be, Mg, Ca, Sr is:

Identify the reaction in which diborane is produced on an industrial scale:

Which one of the following reactions is not feasible?

Consider the following gaseous equilibrium reactions (I), (II) and (III) with equilibrium constants \( K_1 \), \( K_2 \), and \( K_3 \) respectively:
I) \( \frac{1}{2} N_2 + \frac{3}{2} H_2 \rightleftharpoons NH_3 \)
II) \( 2NO \rightleftharpoons N_2 + O_2 \)
III) \( H_2 + \frac{1}{2} O_2 \rightleftharpoons H_2O \)
The correct expression for the equilibrium constant for the gaseous equilibrium reaction: \[ 2NH_3 + \frac{5}{2} O_2 \rightleftharpoons 2NO + 3H_2O \]

The de Broglie wavelengths of two fast-moving particles \( X \), \( Y \) are \( 1 \) nm, \( 3 \) nm respectively. Mass of \( X \) is nine times the mass of \( Y \). The ratio of kinetic energies of \( X \), \( Y \) is:

A molecule has a T-shape. The total number of electron pairs in the valence shell of its central atom is:

The sum of bond order values of \( C_2 \) and \( O_2^{2-} \) is \( x \), which is equal to the sum of bond order values of \( a, b, \) and \( c \). What are \( a, b, \) and \( c \)?

If the maximum and minimum amplitudes of a modulated wave are \( 25 \) V and \( 5 \) V respectively, then the modulation index is:

The ratio of the magnitudes of the electric field and \( 10^8 \) times the magnetic field of a plane electromagnetic wave is:

In Young's double slit experiment, the distance between the slits is \( 0.2 \) cm, the distance between the screen and the slits is \( 1 \) m. If the wavelength of light used in the experiment is \( 5000 \) Å, then the distance between two consecutive dark fringes on the screen is:

A particle of mass \( 0.2 \) g and charge \( 2 \) C is released from rest in a uniform electric field of \( 20 \) N/C. The kinetic energy of the particle after moving a distance of \( 20 \) cm is:

The capacitance of a spherical capacitor is \( 100 \) pF. If the spacing between the two spheres is \( 1 \) cm, then the radius of the inner sphere of the capacitor is:

When a wire is connected in the left gap of a metre bridge, the balancing point is at \( 40 \) cm from the left end of the bridge wire. If the wire in the left gap is stretched so that its length is doubled and again connected in the same gap, then the balancing point from the left end of the bridge wire is:

In a Carnot engine, if the work done during isothermal expansion is 25% more than the work done during isothermal compression, then the efficiency of the engine is:

If the given graph shows the logarithmic values of pressure (P) and volume (V) of an ideal gas, then the ratio of the specific heat capacities of the gas is:

The internal energy of one mole of a rigid diatomic gas at absolute temperature \( T \) is: